## Message 17480 from Yahoo.Groups.Primenumbers

Return-Path: <kermit@...> X-Sender: kermit@... X-Apparently-To: primenumbers@yahoogroups.com Received: (qmail 10242 invoked from network); 3 Jan 2006 04:33:13 -0000 Received: from unknown (66.218.66.166) by m26.grp.scd.yahoo.com with QMQP; 3 Jan 2006 04:33:13 -0000 Received: from unknown (HELO mail.bortnet.com) (64.151.110.218) by mta5.grp.scd.yahoo.com with SMTP; 3 Jan 2006 04:33:13 -0000 Received: from your-4105e587b6 (fl-71-0-163-237.dyn.sprint-hsd.net [71.0.163.237]) by mail.bortnet.com (Postfix) with ESMTP id 400E161E4; Mon, 2 Jan 2006 19:58:15 -0800 (PST) MIME-Version: 1.0 Message-Id: <43B9FE72.000003.01044@YOUR-4105E587B6> Date: Mon, 2 Jan 2006 23:32:50 -0500 (Eastern Standard Time) X-Mailer: IncrediMail (4001930) References: <001b01c60e2b$16daabd0$16dc52c3@jensathlonxp> To: "Prime Numbers" <primenumbers@yahoogroups.com>, "Jens Kruse Andersen" <jens.k.a@...> X-FID: PLAINTXT-NONE-0000-0000-000000000000 X-Priority: 3 X-Originating-IP: 64.151.110.218 X-eGroups-Msg-Info: 1:12:0:0 From: "Kermit Rose" <kermit@...> Reply-To: "Kermit Rose" <kermit@...> Subject: Re: [PrimeNumbers] Prime GAP of 364,188 X-Yahoo-Group-Post: member; u=655161; y=TgsK_u7Rblr6cwx--dZJEaIj0yNGlrtSGvrdoX429CkZBhN3oQ X-Yahoo-Profile: kermit1941 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printableFrom: Jens Kruse Andersen Date: 12/31/05 12:02:34 To: Prime Numbers Subject: Re: [PrimeNumbers] Prime GAP of 364,188 =20 Jose Ramòn Brox wrote: > > There are better methods to choose which numbers have small factors. > > Such methods were used for the 2 known Megagaps: > > http://hjem.get2net.dk/jka/math/primegaps/megagap.htm > > Indeed a very intelligent approach: I was wondering how all that records > could be arrived at, and now I know. Are there any other approaches, > or yours is the main used one? AFAIK, all efficient searches for large gaps between large primes/prp's (e.g above 50 digits) have included some variant of these points: ******** From Kermit kermit@... Has anyone tried this approach? To product a prime gap choose prime divisors, p1, p2, p3, .... pN. Lets pick p1 = 2, p2 = 3, p3 = 5, etc. Choose k = -1 mod 2. This makes k odd. choose k = -2 mod 3. Now we don't need to worry about -3, -5, -7, etc because these are all taken care of by choosing k = -1 mod 2. choose k = - 4 mod 5. Now we don't need to worry about -5, - 8, - 11, - 14, etc, because these are already taken care of by k = -2 mod 3. choose k = -6 mod 7. Now we don't need to worry about -9, -14, -19, - 24, etc because these are taken care of by k = -4 mod 5. Choose k = - 10 mod 11. Now we don't need to worry about -6, -13, -20, etc because these are taken care of by=20 k = -6 mod 7. etc. So with the 5 primes 2,3,5,7,11 we can assure a prime gape of length 11. Each prime we add to the list lengthens the prime gap by more than 1. In fact, note that we just defined k = 1 mod 2, k = 1 mod 3, k = 1 mod 5, k = 1 mod 7. So the minimum k satisfying this is k = 2 * 3 * 5 * 7 * 11 + 1 = 2311. In fact, I just realized why the pattern I chose will always yield p#+1 as the value of k. Thus I just proved that the prime gap of p# is at least p. Perhaps some other way of assigning the negative values will produce a sieve of higher merit. [Non-text portions of this message have been removed]Message 17459 Message 17461 Message 17462 Message 17488